## MULTIPLYING POLYNOMIALS

In this lesson on Multiplying Polynomials, we will learn:

• Multiplying monomials
• Multiplying binomials briefly and
• Multiplying polynomials

• Pre-requisites for understanding this multiplying polynomials lesson:

The following rules for multiplying positive and negative numbers:

Multiplication Rule 1:

–  × + = –

Example 1:

– a × +b = – ab

Multiplication Rule 2:

× –   = +

Example 2:

• a × – b  = +ab

Multiplication Rule 3:

+  × –  = – ab

Example 3:

+ a × –  b= – ab

Multiplication Rule 4:

+  × +  = +

Example 4:

+a  × + b = + ab

Multiplying polynomials is one of the four fundamental operations of algebraic expressions.

The other three fundamental operations on algebraic expressions are:

• Subtraction of polynomials
• Division of polynomials

Now, let us learn Multiplying Polynomials by remembering the following multiplication rules (for multiplying polynomials)

Step 1: Multiply the Numerical Coefficients separately

Step 2: Multiply the literal coefficients separately and

Step 3: Multiply the above two products (the numerical coefficients product and the literal coefficients product)

SOLVED PROBLEM NO.1:

Multiply the following two monomials:

3a2 and 5a3

Step 1: Numerical coefficients product = 3 × 5 = 15

Step 2: literal coefficients product = a2 × a3 = a5 (from the law of exponents – if bases are same, then add the powers)

Step 3:

Now, write the product of the numerical coefficients (15) and literal coefficients (a5), i.e.

15 × a5 = 15a5.

Note:

15a5 stands for the product of 15 and a5, just as

pq means product of p and q i.e. p × q, and

3a means 3 × a.

Now, Multiplying a Monomial with a Binomial

SOLVED PROBLEM NO 2:

Multiply  the Monomial 3a2 with the binomial x + y

Solution:

First set up the product as below:

3a2 × (x + y)

Now, Recall the distributive property a (b + c) which is:

a(b + c) = ab + ac

So, by applying the above distributive property to 3a2 × (x + y), we get

3a2x + 3a2y.

SOLVED PROBLEM NO 3:

Perform the following multiplication of polynomials:

3a2 (5a3b + 6ac)

Solution:

From the distributive property a (b + c) = ab + ac,

We get the following two terms:

3a2 × 5a3b + 3a2 × 6ac

Now from exponents rules, add powers of same bases and applying the following steps:

Step 1: Multiply the Numerical Coefficients separately

Step 2: Multiply the literal coefficients separately and

We get:

3a2 × 5a3b = 3×5×a2 ×a3 × b   = 15 a5×b and

3a2 × 6ac = 3 × 6 × a2 × a × c = 18a3c, so finally the product of

3a2 and (5a3b + 6ac) is 15 a5×b + 18a3c.

MATH SKILLS RELATED TO MULTIPLYING POLYNOMIALS

• Worksheets on Multiplying polynomials

• Worksheets on Multiplying binomials

• Multiplying Fractions

• Multiplying Numbers

• Multiplying Decimals

• Multiplying Exponents

• Multiplying Fractions

• Multiplying Matrices

• Multiplying Mixed Fractions

• Multiplying Polynomials